ω
/2
π
(GHz)

Δ
V
(V)

λ
/2
π
(MHz)

J
/2
π
(MHz)


2.5

10

1.2

0.7

2.5

20

2.3

2.7

3.5

10

1.2

0.6

3.5

20

2.5

2.3

 The nanoresonator is assumed to be aluminum, with the following dimensions: length L = 0.70 μm (0.6 μm) for the 2.5 GHz (3.5 GHz) mode; width w = 45 nm; thickness t = 50 nm; The transmonNR coupling λ was calculated from circuit theory, assuming the systems to be in resonance, and is given by \(\lambda= \omega\sqrt{\frac{dC}{dx}\frac{C \Delta V ^{2}}{m \omega ^{2} d C_{Q}}}\). Here C = 20aF, dC/dx = 6 × 10^{−11} F/m are the coupling capacitance and its first derivative respectively, which are estimated from finite element simulations assuming a gap of d = 20 nm. \(C_{Q}=50 fF\) is the transmon shunt capacitance, which would yield a charging energy of \(E_{C}/h = 400\) MHz. Finally, m = 0.52. ρwtL is the effective mass of the third mode, where ρ is the density of aluminum. The same parameter values are assumed for the calculation of J.